Their advocacy group, Mayors Against Illegal Guns, has pushed for doing buyer background checks at gun shows since it began in 2006. The group didn’t have a position on military-style weapons until it became clear that an assault-weapons ban would be part of the Senate debate. Only then did it decide to support it.

Now the conversation on Capitol Hill is right where Bloomberg, Menino, and other gun-control advocates have long wanted it. Lawmakers are working hard to figure out how to best expand background checks and keep better records on felons and people with mental illnesses, who are banned from buying firearms.

It’s the kind of back-and-forth that one would expect from serious policymakers grappling with a difficult problem, not a Congress mired in deadlock over budgets and other issues for years. But Republicans and Democrats aren’t even bothering to trade political barbs on whether it makes sense to outlaw assault weapons, because they know that proposal is going nowhere.

The hardest part of the broader debate on gun-control happens Thursday, when sponsors of several different gun-related proposals will find out if they can overcome a Senate filibuster with 60 votes. It’s looking like they will succeed, despite dogged protests from Sens. Mike Lee, R-Utah, and Ted Cruz, R-Texas, who have not backed off from their threats to block the legislation.

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Expanded background checks are receiving a lot of attention from Republicans and Democrats, an indication that the debate has already moved past the filibuster stage. “I’m encouraged and excited that all signs are right now that we’ll have a vote on a final bill,” said Sen. Jeff Flake, R-Ariz. Flake has not committed to how he will vote on the bill itself, but he does not support Lee and Cruz’s filibuster attempt.

Sens. Joe Manchin, D-W.Va., and Pat Toomey, R-Pa., put forward a compromise on background checks Wednesday that would require the checks at gun shows and for Internet sales, but not for “over-the-fence” firearms sales from person to person. The measure is designed to replace universal-background-check language sponsored by Sen. Chuck Schumer, D-N.Y., that does not have enough support to pass the Senate.

Manchin and Toomey’s proposal may go a long way toward easing gun owners’ fears that they won’t be able to give or sell weapons to family and friends, but it still might not assuage worries that a national gun registry could be in the works. Some Republicans have said they will only support expanded background checks if there are no records of the private gun purchases. Manchin and Toomey’s bill would not eliminate firearms records, but it would explicitly prohibit the federal government from cataloging them and would impose stiff penalties for misuse.

“I don’t know if it will belie peoples’ concerns,” said Sen. Lindsey Graham, R-S.C., of the record-keeping language.

But Sen. Susan Collins, R-Maine, said she was encouraged by the compromise proposal because gun-purchase records would be kept by dealers, not federal enforcers. “I’m not trying to dismantle the record keeping that is done by federal firearms dealers,” Collins said. “I believe they are headed in the right direction. It’s a very promising approach.”

Getting rid of the gun-purchase records would damage law enforcement’s ability to trace weapons that are used in crimes, according to Richard Feldman, president of the Independent Firearm Owners Association, a gun-owners group that supports background checks at gun shows.

“If law enforcement knows that a gun was used at a crime scene in Boston, they’ll be able to trace the gun to me [in New Hampshire]. I can invite them in … and tell them my gun was stolen one week ago. Now they have some information about who is misusing the gun,” Feldman said.

Republicans are preparing an alternative gun proposal to counter the legislation offered by Senate Majority Leader Harry Reid. It will include a variety of measures, including updates to mental-health reporting requirements for states; increased penalties on gun trafficking; clarification that states will not violate health privacy laws if they report certain mental-health cases to a national database; and school-safety provisions, according to Graham.

Reid has promised Manchin and Toomey that their proposal on expanded background checks will be the first amendment to receive a vote.